Optimal. Leaf size=110 \[ \frac {1}{7} x^7 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b x^6 \sqrt {\frac {1}{c^2 x^2}+1}}{42 c}-\frac {5 b \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{112 c^7}+\frac {5 b x^2 \sqrt {\frac {1}{c^2 x^2}+1}}{112 c^5}-\frac {5 b x^4 \sqrt {\frac {1}{c^2 x^2}+1}}{168 c^3} \]
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Rubi [A] time = 0.06, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6284, 266, 51, 63, 208} \[ \frac {1}{7} x^7 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b x^6 \sqrt {\frac {1}{c^2 x^2}+1}}{42 c}-\frac {5 b x^4 \sqrt {\frac {1}{c^2 x^2}+1}}{168 c^3}+\frac {5 b x^2 \sqrt {\frac {1}{c^2 x^2}+1}}{112 c^5}-\frac {5 b \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{112 c^7} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rule 6284
Rubi steps
\begin {align*} \int x^6 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {1}{7} x^7 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b \int \frac {x^5}{\sqrt {1+\frac {1}{c^2 x^2}}} \, dx}{7 c}\\ &=\frac {1}{7} x^7 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {1+\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{14 c}\\ &=\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {(5 b) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1+\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{84 c^3}\\ &=-\frac {5 b \sqrt {1+\frac {1}{c^2 x^2}} x^4}{168 c^3}+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(5 b) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{112 c^5}\\ &=\frac {5 b \sqrt {1+\frac {1}{c^2 x^2}} x^2}{112 c^5}-\frac {5 b \sqrt {1+\frac {1}{c^2 x^2}} x^4}{168 c^3}+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {(5 b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{224 c^7}\\ &=\frac {5 b \sqrt {1+\frac {1}{c^2 x^2}} x^2}{112 c^5}-\frac {5 b \sqrt {1+\frac {1}{c^2 x^2}} x^4}{168 c^3}+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {(5 b) \operatorname {Subst}\left (\int \frac {1}{-c^2+c^2 x^2} \, dx,x,\sqrt {1+\frac {1}{c^2 x^2}}\right )}{112 c^5}\\ &=\frac {5 b \sqrt {1+\frac {1}{c^2 x^2}} x^2}{112 c^5}-\frac {5 b \sqrt {1+\frac {1}{c^2 x^2}} x^4}{168 c^3}+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^6}{42 c}+\frac {1}{7} x^7 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {5 b \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{112 c^7}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 107, normalized size = 0.97 \[ \frac {a x^7}{7}-\frac {5 b \log \left (x \left (\sqrt {\frac {c^2 x^2+1}{c^2 x^2}}+1\right )\right )}{112 c^7}+b \sqrt {\frac {c^2 x^2+1}{c^2 x^2}} \left (\frac {5 x^2}{112 c^5}-\frac {5 x^4}{168 c^3}+\frac {x^6}{42 c}\right )+\frac {1}{7} b x^7 \text {csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 208, normalized size = 1.89 \[ \frac {48 \, a c^{7} x^{7} + 48 \, b c^{7} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 48 \, b c^{7} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 15 \, b \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) + 48 \, {\left (b c^{7} x^{7} - b c^{7}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (8 \, b c^{6} x^{6} - 10 \, b c^{4} x^{4} + 15 \, b c^{2} x^{2}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{336 \, c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 127, normalized size = 1.15 \[ \frac {\frac {c^{7} x^{7} a}{7}+b \left (\frac {c^{7} x^{7} \mathrm {arccsch}\left (c x \right )}{7}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (8 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}-10 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+15 c x \sqrt {c^{2} x^{2}+1}-15 \arcsinh \left (c x \right )\right )}{336 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 158, normalized size = 1.44 \[ \frac {1}{7} \, a x^{7} + \frac {1}{672} \, {\left (96 \, x^{7} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, {\left (15 \, {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{3} - 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{6}} - \frac {15 \, \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} + \frac {15 \, \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^6\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{6} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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